Strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains

TL;DR

This paper proves the existence and stability of strong solutions to the Keller-Segel-Navier-Stokes system in bounded Lipschitz domains, using advanced regularity techniques, and shows solutions are globally bounded for smoother data.

Contribution

It establishes local and global strong solutions for the coupled system in Lipschitz domains with small initial data, and demonstrates exponential stability of equilibria, advancing the mathematical understanding of chemotaxis-fluid models.

Findings

Existence of local and global strong solutions for small data.

Exponential stability of non-trivial equilibria.

Global boundedness and positivity preservation for smoother data.

Abstract

Consider the coupled Keller-Segel-Navier-Stokes or the chemotaxis-consumption-Navier-Stokes system in bounded Lipschitz domains for general coupling terms which, e.g., include buoyancy forces. It is shown that these systems admit local strong as well as global strong solutions for small data in the setting of critical Besov spaces. Moreover, non-trivial equilibria are shown to be exponentially stable. For smoother data, these solutions are shown to be globally bounded and to preserve positivity properties. The approach presented is based on optimal $\mathrm{L}^q$-regularity properties of the Neumann Laplacian and the Stokes operator in bounded Lipschitz domains.